Two-mode oscillator Casimir identity ($H^2=4{\omega }^2{J}^2+{\hslash }^2{\omega }^2$)

For the two-dimensional isotropic harmonic oscillator, $H=\hslash \omega (N+1)$ and the Schwinger boson representation gives $J^2={\hslash }^{2}N(N+2)/4$. Substitution yields the displayed relation, including its quantum constant term. The fixed-total-occupation sector is a spin-$j=N/2$ representation, so energy is $\hslash \omega (2j+1)$ with degeneracy $2j+1$.